The use of curved lines (arches, curved beams) and curved surfaces (shells, vaults, domes, membranes) in architecture arises out of several needs. There is the pragmatic need for the efficient use of material to cover space, an idea that becomes increasingly relevant with depleting resources. This economy of material can translate into decreased costs of building. There is the architectural need for "comfort" in inhabiting spaces and structures that are "organic" and mirror the constructions in nature. There is the philosophical need for living in harmony with nature. For these reasons, curved space structures are desirable in architecture.
Curved space structures are characterized by curved surfaces and curved lines. The curved surfaces can be single-curved as in cones and cylinders, or doubly-curved as in spheres and saddles. Architectural structures based on singly-curved and doubly-curved surfaces are well-known. In either case, the surfaces can be continuously smooth surfaces as in cast shells made of concrete or plastics, or tensile membranes made of reinforced nylon fabrics. Alternatively, the curved surfaces can be decomposed into polygonal areas which can be manufactured separately as parts of the structure and the entire surface assembled out of these pre-made parts. Such space structures have relied upon a geometric subdivision of the surface into polygonal areas. In all prior art, such geometric subdivision is based on periodic subdivision of the fundamental region of the structure; the fundamental region is the minimum spatial unit of the structure from which the entire structure can be generated using symmetry operations of reflection, rotation, translation and their combinations. In addition, the prior art of modular space structures has retained the global symmetry of the space structure.
In contrast to the prior works, this application discloses three new classes of curved space structures not taught by the prior art of building. One class comprises globally symmetric space structures where the fundamental region is subdivided into rhombii in a non-periodic manner. The second class where the entire polygonal faces of symmetric space structures are subdivided non-periodically or asymmetrically into rhombii and the structure retains only partial global symmetry or is completely asymmetric. The third class of structures are those in which the rhombii of non-periodic subdivisions are subdivided further in a periodic manner.
The structual advantages of the "new" space structures disclosed here remain to be examined and analyzed. But as the history of building art reveals, new geometries have always led to special architectural, structural, functional, or aesthetic advantages. The aesthetic appeal of non-periodic space structures cannot be overemphasized as these are a marked departure from the conventional space structures which, with recent exceptions, have relied upon periodicity as a device to cover space and span structures. Curved space structures with non-periodic subdivisions are new and are likely to advance the building art of the future.
Prior art includes U.S. Pat. No. 4,133,152 to Penrose which discloses the Penrose tiling, U.S. Pat. No. 5,007,220 to Lalvani which discloses prismatic nodes for periodic and non-periodic space frames and related tilings, U.S. Pat. No. 5,036,635 to Lalvani which discloses periodic and non-periodic curved space structures derived from vector-stars, U.S. Pat. No. 3,722,153 to Baer which discloses nodes of icosahedral symmetry for space frames, the work of T. Robbin which suggests the use of dodecahedral nodes for "quasicrystal" space structures using the De Bruijn method, the work of K. Miyazaki which discloses the 3-dimensional analog of the Penrose tiling. Prior work also includes known plane-faced zonohedra having tetrahedral, octahedral and icosahedral symmetry and derived from corresponding symmetric stars published in H. S. M. Coxeter's Regular Polytopes (Dover, 1973). Other related publications include Lalvani's article `Continuous Transformations of Non-Periodic Tilings and Space-Fillings` in Fivefold Symmetry by I. Hargittal (World Scientific, Singapore, 1992), and citations to Lalvani in J. Kappraff's Connections: The Geometric Bridge Between Art and Science (McGraw-Hill, 1991, p. 246-249).
None of the prior art deals with non-periodic subdivisions of the fundamental region of various symmetric space structures, nor does it deal with non-periodic and asymmetric subdivisions of the surfaces of space structures. Further, prior art does no deal with the non-peridic subdivision of architecturally useful curved space structures like domes, vaults and related structures. Going further, the prior art does not teach such subdivisions for higher-dimensional and hyperbolic space structures.